


Convergence

by Bitenomnom



Series: Mathematical Proof [10]
Category: Sherlock (TV)
Genre: BDSM, Bondage, M/M, Mathematics, PWP, just giving fair warning, seriously it looks like it's 700 words long but half of that is a mathematical proof so, very very short
Language: English
Status: Completed
Published: 2012-09-19
Updated: 2012-09-19
Packaged: 2017-11-14 14:27:49
Rating: Explicit
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 704
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/516184
Author URL: https://archiveofourown.org/users/Bitenomnom/pseuds/Bitenomnom
Summary: <blockquote class="userstuff">
              <p>“Look at you,” Sherlock chuckled.  John, of course, could not look at himself. <i>Situate in front of mirror next time,</i> Sherlock made a mental note. “Heart rate increasing. Back arched.” Not that John had much of a choice in that matter, either.</p>
            </blockquote>





	Convergence

**Author's Note:**

> Okay, I promise I'm not just doing a tiny little PWP just to get your hopes up and then destroy them when you realize the actual thing isn't even 300 words long. I just need to get to bed about half an hour ago...I was busy today, and then spent some time devising [a new way to tie these shoes I have](http://imageshack.us/a/img201/1797/shoesh.jpg) because I didn't want to cut the leather laces. So knots were on my mind.
> 
> Now that there are so many people reading these compared to what I expected, I'm feeling the pressure, ahaha. So, um, just a reminder that this is a daily exercise for me, so some days are going to be better than others and I'm sorry that they're not all great. @_@ But also thank you to all for being so awesome and reading this series and enjoying it and commenting and everything. I'm constantly amazed.

**Convergence**

Theorem: Let f be a function in the real numbers such that f’(x) > 0 (that is, f is increasing) and f’’(x) > 0 (f is concave up), and f(x) has a root r. Then r is unique and for every x0 in the real numbers, Newton’s Method converges to r quadratically. (This means that if you are using Newton’s Method to estimate the value of a function with these properties, you will get a pretty close approximation very quickly compared to, for instance, the Bisection Method.)

 

Proof: Previously established that en+1 = f’’(zn) en2 / 2 f’(xn), where en = xn – r (the difference between our current estimate and the actual root). Essentially, the difference at the next iteration depends on the square of the previous one, as well as the second derivative at some closer point than the previous distance and the first derivative at the previous distance. Therefore, en+1 > 0 since both the numerator and denominator are positive as a part of the requirements established in the theorem. For all n, xn+1 > r (your estimate is always going to be bigger than the actual root). Because of this and the fact that the function is increasing, f(xn) > f(r) = 0. [f(r) is the root so it equals zero.] Therefore, f(xn) > 0 (the function is always positive).

 

Now, xn+1 = xn – f(xn)/f’(xn) < xn so {xn} (the sequence of xn’s) is decreasing.

Since en+1 = en – f(xn)/f’(xn) <en we also have {en} is decreasing.

 

Since {xn} and {en} are decreasing and bounded below by r and 0 respectively, then {xn} and {en} converge.

 

Now, assume xn converges to some P. Then xn+1 goes to P and xn goes to P, so

 xn+1 = xn – f(xn)/f’(xn) becomes P = P – f(P)/f’(P). So f(P)/f’(P) = 0. Therefore f(P) = 0 and P is a root, r.

 

Finally, r is unique because if we assume the opposite, that f(r1) = f(r2) = … = 0 (there are multiple roots), then we get by Rolle’s Theorem that there’s some c between two of these r values such that f’(c) = 0, which is a contradiction because we already assumed that f’(x) > 0 for all x values. Therefore, it must be the case that r1 = r2 = … = r.

 

 

***

 

            “Look at you,” Sherlock chuckled.

            John, of course, could not look at himself. _Situate in front of mirror next time,_ Sherlock made a mental note.

            “Heart rate increasing. Back arched.” Not that John had much of a choice in that matter, either; his feet were tied and his legs drawn up behind him, the binding connected to the bedposts and ceiling. (Sherlock would slip a bit extra in with the rent this month and let Mrs. Hudson discover why later.) His hands were fixed in front of him, forcing his back into a curve.

            John grunted. There wasn’t much else he could do, in his state.

            “And, let’s not forget, _bound_.”

            “Rrgh,” said John.

            “I see only one way this can end. Do you agree?”

            “Mmrmph.”

            “Of course you do.” Sherlock slid one hand up the inside of John’s thigh until his fingers brushed his perineum. From above John he tried to memorize the curve of John’s arse—but there were so many details. Additional trials would be necessary, to refine and refine and refine the image. Sherlock licked his fingers and painted teasing circles around John’s arsehole, and John struggled against the ropes in vain to push it up toward Sherlock. “Tsk,” Sherlock tutted. “So impatient.”

            John thrashed—what little thrashing he could do—and, Sherlock was certain, was swearing against the gag. “Fffmfmph,” he said.

            “That’s right,” he made no attempts to rid condescension from his voice. Sherlock sat back on his haunches momentarily to reposition himself, digging his fingers into John’s arse as he pulled his cheeks apart and lined his erection up with his hole. “Convergence.”


End file.
